Many mathematical models of real world phenomena take the form of partial differential equations which may exhibit singularities in finite time. In this talk, I shall discuss adaptive strategies in both space and time for the resolution of such behaviour. This adaptive approach is based on scaling properties of the PDE, which leads to uniform error estimates, rather than on information about particular solutions. Examples in one and two dimensions from a variety of applications will be presented.

This is joint work with C. J. Budd (University of Bath).